Structural Stability of Ruthenium Nanoparticles: A Density Functional

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Structural Stability of Ruthenium Nanoparticles: A Density Functional Theory Study Yusuke Nanba, Takayoshi Ishimoto, and Michihisa Koyama J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b08672 • Publication Date (Web): 22 Nov 2017 Downloaded from http://pubs.acs.org on November 23, 2017

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The Journal of Physical Chemistry

Structural Stability of Ruthenium Nanoparticles: A Density Functional Theory Study Yusuke Nanba*†, Takayoshi Ishimoto‡, Michihisa Koyama*†,‡ †

INAMORI Frontier Research Center, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan ‡

Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, HigashiHiroshima, Hiroshima 739-8527, Japan [email protected], [email protected]

ACS Paragon Plus Environment

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ABSTRACT We have analyzed the crucial factors that stabilize face-centered cubic (fcc) ruthenium nanoparticles (Ru-NPs) using the density functional theory method. We calculated the cohesive energy of the decahedral fcc, icosahedral fcc, truncated octahedral fcc, and hexagonal close packed (hcp) Ru-NPs with 55 and 1557 atoms. The cohesive energy of the icosahedral fcc Ru-NPs became closer to that of the hcp Ru-NPs with decreasing number of atoms, i.e., particle size. This characteristic is mainly caused by the high coordination number of the icosahedral fcc Ru-NP and the negative twin boundary energy for fcc {111}. On the other hand, the d-band center of Ru atoms in the surface layer of icosahedral fcc Ru-NPs is less negative than those of the other structures. This characteristic is caused by the longer interatomic distance between Ru atoms in the surface layer of the icosahedral fcc Ru-NP. Together with the structural stability, the icosahedral fcc structure shows a unique electronic structure compared with the other structures. Our results are expected to be helpful for controlling and designing the properties, such as stability and catalytic activity, of Ru-NPs from the shape of the NP.


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INTRODUCTION Metal nanoparticles (NPs) have different characteristics from the bulk state because of the high surface area-to-volume ratio, quantum size effect, and so on.1 Typical examples are the particle size dependence of the catalytic activities in Pt- and Au-NPs.2−6 In core−shell structured nanoalloys, the properties are different from those of core or shell metal NPs and a mixture of them.7−10 Solid-solution nanoalloys can be synthesized even if they are not immiscible in the bulk state, and also show different properties from the monometal NPs.11−13 In addition, the crystal structure control of monometal NPs14−17 is one of the approaches to change the properties and create new functions of NPs. Recently, ruthenium nanoparticles (Ru-NPs) with face-centered cubic (fcc) structures were reported,18−21 although the bulk Ru monometal shows the hcp structure and maintains the hcp structure at high pressure22 and high temperature.23 The synthesis of monometallic fcc and hcp structures of Ru-NPs is controlled by the solvent and Ru precursors.18−21 Moreover, nanoalloys and nanoframes having the Ru fcc surface were synthesized with the aid of other fcc transition metals such as Pd and Pt.24−26 The fcc and hcp structures of the Ru-NPs affect the catalytic activity. While the hcp Ru-NPs are more reactive in some catalytic activities, fcc Ru-NPs are superior in other catalytic activities, such as hydrogenation of styrene and 4nitrochlorobenzene.25 One of the most interesting features of the fcc and hcp Ru-NPs is the different particle size dependence of the catalytic activity for CO oxidation.18 The hcp RuNPs became more active with decreasing particle size. By contrast, the fcc Ru-NPs showed a different trend compared with hcp Ru-NPs. The crossing point of catalytic activity between the fcc and hcp Ru-NPs is a particle size of about 3 nm. If the stabilization factor of the unstable structure in the bulk state is found, it is expected that NPs having new properties and functions will be created. It is important to understand the size and shape dependence of NPs with different crystal structures.


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So far, the electronic states of NPs consisting of from 10 to several hundred atoms have been analyzed using first-principle calculations. For example, the cohesive energies of Cu-, Pd-, Ag-, Pt-, and Au-NPs were calculated to evaluate their stability.27−35 It was reported that the average interatomic distance shortens with decreasing particle size, while the cohesive energy increases. The values in the bulk obtained by the scaling technique based on the cohesive energy of metal cluster or nanoparticle models are close to those by the bulk calculation. Analysis of the cohesive energy has already been applied to Ru-NPs. The calculated results of Ru clusters up to 64 atoms indicate that the simple cubic is the most stable structure for clusters up to 40 atoms,36 while the icosahedral structure is the most stable for more than 40 atoms. The scaling behaviors for hcp and fcc Ru-NP between 55 and 323 atomic systems were also established by Soini et al.37 The cohesive energies of hcp Ru-NPs were always higher than those of fcc Ru-NPs. In this analysis, only two samples were treated as the icosahedral structure. It is known that different surfaces and shapes appear in NPs from the synthesis conditions even if the fcc structure is shown. The surface of an NP with the fcc structure shows {001} facet or {111} facets, which form a wide variety of shapes, such as decahedron, icosahedron, cuboctahedron, octahedron, and tetrahedron as the fcc NP structures having {001} and/or {111} facets.38−50 Thus, it is necessary to consider multiple shapes of Ru-NPs to discuss the structural stability of Ru-NPs. During the last decade, largescale computing has enabled us to perform first-principles calculations of NPs with the observed particle size.51−53 Challenging by a first-principles calculation for a realistic particle size of NPs is necessary to evaluate the structural stability. In this study, we analyzed the stability of fcc and hcp Ru-NPs with various particle sizes and shapes using density functional theory (DFT). The size and shape dependence of the geometric and electronic structures of Ru-NPs are also discussed.


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METHODS Computational Details. The DFT calculations for the Ru-NP were conducted using the Vienna Ab Initio Simulation Package (VASP) code54,55 with the Perdew–Burke–Ernzerhof56 exchange-correlation functional based on the generalized gradient approximation. A projector augmented wave was used as the interaction between valence electrons and core electrons.57,58 The cutoff energy for the plane wave was 400 eV and the Monkhorst–Pack kpoint grid was 1 × 1 × 1. The convergence of the self-consistent field and the geometry optimization were 1.0 × 10–5 and 1.0 × 10–4 eV/atom, respectively. The vacuum space in the unit cell was set to be more than 12 Å to avoid the interaction with the neighboring NPs. The Bader approach59 was used to calculate atomic charges of Ru atoms in NP from the VASP results. Models of Ru-NPs. Various shapes, such as decahedral, icosahedral, and truncated octahedral structures, have been observed in Pd-, Ag-, Pt-, and Au-NPs as the fcc structure.38−50 We prepared the decahedral and icosahedral structures having only {111} facets and the truncated octahedral structure having both the {001} and {111} facets, as shown in Figures 1 and S1. The decahedral and icosahedral structures were observed for the fcc Ru-NP using high-resolution transmission electron microscopy (HRTEM).18 The truncated octahedral fcc structure was observed in nanoalloys and nanoframes with the Ru fcc surface.24−26 HRTEM also showed that hcp Ru-NP was a truncated hexagonal bipyramid, as shown in Figure 1.

RESULTS AND DISCUSSION Shape Characteristics. Characteristics of the NPs, such as the ratio of surface atoms and average coordination number (CN), depend on decahedral fcc, icosahedral fcc, truncated octahedral fcc, and hcp Ru-NP (Table S1). The ratio of surface atoms in decahedral fcc Ru-


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NP is very different from other types of Ru-NPs. Even when the particle size of the decahedral fcc Ru-NP is large, the fraction of surface atoms is more than 0.45 (Figure S2). In addition, the ratio of atoms in the surface layer to that in the subsurface layer of the decahedral fcc Ru-NP is more than 2.0 (Table S1). On the other hand, the average CN of the icosahedral fcc Ru-NP is larger than those of other types when the particle size is similar. Analysis of the Structural Stability Using the Cohesive Energy. We discuss the stability of Ru-NP based on the cohesive energy. The cohesive energy per atom is expressed as ε coh = (ε N − N ∗ ε1 ) N ,

(1)

where εN and ε1 represent the total energy of the Ru-NP and the ground-state energy of the Ru atom, respectively. The ground-state energy of the Ru atom was the quintuplet configuration 4d75s1. Note that the sign of the cohesive energy is different from that in a previous study.37 The cohesive energy is proportional to N–1/3, where N represents the number of atoms in the NP.27−35 Figure 2 shows the cohesive energies for each structure as a function of N–1/3, where the specific values of cohesive energies are shown in Table S2. The broken lines represent the linear regression lines for each structure. The hcp Ru-NP was stable in the large N region. This result is consistent with the Ru metal exhibiting the hcp structure in the bulk state (Table S3). The linear regression lines of the decahedral and truncated octahedral fcc Ru-NPs did not intersect with those of the icosahedral fcc and hcp lines. The decahedral and truncated octahedral fcc Ru-NPs were calculated to be unstable in the same sizes of NP. The linear regression line of the icosahedral fcc Ru-NP intersected with that of hcp Ru-NP at N = 103. This result indicates that the icosahedral fcc Ru-NP becomes stable compared with other RhNPs when the particle size becomes small. We evaluated the influence of surface formation to understand the variations of cohesive energies of Ru-NPs. As shown in eq S4, the cohesive energy of NP is affected by the surface energy. Table 1 shows the surface energies of Ru-NPs for each structure. The surface energy


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of the hcp Ru-NPs was the highest in Ru-NPs at similar size. In the bulk state, the cohesive energy of hcp Ru is lower than that of fcc Ru (Table S3), which causes the larger energy loss of bond breaking for hcp Ru surface formation. As the particle sizes of Ru-NPs become small, the energy loss for surface formation gives a larger influence on the stability of Ru-NP. Another characteristic is the low surface energy of the decahedral fcc Ru-NP. The decahedral fcc Ru-NPs show large surface ratio compared with those of other Ru-NPs (Figure S2), in other words, the wider surface area at similar size. The wide surface area leads to the lower surface energy although the cohesive energies of decahedral fcc Ru-NPs were less negative than those of other Ru-NPs. The surface energies of Ru-NPs are sensitive to the surface area. We estimated the surface energies of typical hcp and fcc surfaces to understand the plane orientation dependence of the surface energy. Slab models with seven layers based on the 2 × 2 supercell were used (Figure S4) for the surface energy calculation. Table 1 shows the surface energies of typical hcp and fcc surfaces. The fcc (111) surface exhibited the lowest surface energy in these models. The CN of Ru atoms in the surface layer (Z) is also shown in Table 2. With decreasing Z, the surface energy became larger by the bond breaking from the bulk state. The CN is one of the factors determining the stability of the NP. We took the average of CN from the surface and inner Ru atoms. Figure 3 shows the average CN of RuNP as a function of N–1/3. The average CN of the icosahedral fcc Ru-NP was higher than those of other types of Ru-NPs. The icosahedral fcc Ru-NPs have many atoms of CN = 9 at face and CN = 8 at edge. In addition, the inner atoms are 40-60% in NP (Table S1). Thus, the average CN of icosahedral fcc Ru-NPs are more than 9 except for Ru55. Although the decahedral fcc Ru-NP has only the fcc {111} facet, the average CNs of the decahedral fcc Ru-NPs are similar to those of the truncated octahedral fcc and hcp Ru-NPs because the ratio of surface atoms in decahedral fcc Ru-NP is higher than those in other types of Ru-NPs of similar particle size (Figure S2). This means that many bonds are broken to form the fcc


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{111} facet of the decahedral fcc Ru-NPs. The influence of the surface energy on the cohesive energy is large in the decahedral fcc Ru-NPs with a high ratio of surface atoms. The truncated octahedral Ru-NP is composed of not only fcc {111} but also fcc {100} facets. The CN of Ru atoms in the fcc (100) surface layer is eight and the surface energy of the fcc (100) _

surface was close to that of the hcp (1010) with CN = 8. Because the average CN of the truncated octahedral fcc Ru-NP was much the same as that of hcp Ru-NP, the cohesive energy of the truncated octahedral fcc Ru-NP remained higher than that of hcp Ru-NP. In this study, we considered the decahedral fcc and icosahedral fcc Ru-NPs because both the NPs show only fcc {111} facet. However, the truncated decahedral fcc Ru-NP (Figure S6) shows the more average CN than the decahedral fcc Ru-NPs. The cohesive energies of the truncated decahedral Ru-NPs were lower than that of the decahedral Ru-NP, which is consistent with the relation between CN and cohesive energy. The CN is one of important factors for determining the stability for the fine shape. Thus, Ru-NP exhibits the icosahedral fcc structure with a high CN to decrease the energy loss of the bond breaking. Decahedral and icosahedral fcc NPs are composed of some tetrahedra with fcc {111} facets. The elastic strain and twin boundary are formed from the packing of the tetrahedra. The elastic strain density and twin boundary energy in the unit area affect the stability of the NP.60 These contributions should be considered in evaluating the cohesive energies of the decahedral and icosahedral fcc Ru-NPs. The elastic strain energy density W is expressed in terms of elastic constants C11, C12, and C44 (eqs S6 and S7). These elastic constants were estimated using the bulk model for the fcc structure (Table S4). The obtained elastic strain energy densities W of the decahedral and icosahedral fcc structures were 3.24 × 10–4 and 6.39 × 10–3 eV/Å3, respectively. These values were larger than those of Pd in Reference 60 (1.31 × 10–4 and 1.80 × 10–3 eV/Å3). The contributions to the cohesive energies of the decahedral and icosahedral fcc Ru-NPs were the order of 10–3 and 10–2 eV/atom, respectively (Tables S5).


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Next, the twin boundary energy per unit area was estimated using the twin boundary models, as shown in Figure 4. We assumed ABCABC-type stacking as in the original fcc {111}. If the twin boundary is located at A, B and C are arranged as a reflection at A. The original fcc {111} stacking model and twin boundary model are denoted as Σ1 and Σ3, respectively. The twin boundary energy per unit area is expressed as γt =

E Σ 3 − E Σ1 , 2A

(2)

where A is the area of the twin boundary. EΣ1 and EΣ3 represent the total energies of the original fcc {111} stacking model and the twin boundary model, respectively. For _

_

comparison, the twin boundary energy per unit area for hcp {1012} and {1011}, which are typical twin boundaries for the hcp structure, were also analyzed (Figures S7 and S8). Table 3 _

_

shows the twin boundary energies per unit area for the fcc {111}, hcp {1012}, and hcp {101 _

_

1}. The obtained twin boundary energies per unit area for hcp {1012} and {1011} were _

positive. Mg and Au also show positive twin boundary energies per unit area for hcp {1012} _

and {1011}.61,62 On the other hand, the obtained twin boundary energy per unit area for fcc {111} was negative. This result is different from Pd and Ag.60 The twin boundary for fcc {111} makes the icosahedral fcc Ru-NPs stable, while it makes the icosahedral fcc Pd- and Ag-NPs unstable. In the twin boundary for the fcc {111}, the local structure near the twin boundary is similar to the hcp {0001} stacking (ABAB…), as shown in Figure 4. The hcp structure is stable in the bulk state of Ru (Table S2). Therefore, the structure similar to the hcp structure included in the fcc {111} model results in the stable energy of the twin boundary fcc model. The twin boundary energy per unit area for fcc {111} became less negative with increasing periodic ABC units. This result suggests that the twin boundary for fcc {111} has a large influence on the stability in the small icosahedral fcc Ru-NP.


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The high CN and negative twin boundary energy play important roles in making the cohesive energy of the icosahedral fcc Ru-NP in the small particle size relatively stable. This study was performed using a model that was based on the shape observed in HRTEM. Therefore, the hcp Ru-NPs employed in this study are long in the [0001] direction. The surface and CN of NP depend on the truncation length. The specified truncation length results in lower cohesive energy. In Reference 37, hcp Ru-NPs consisting of 57 and 153 atoms (Figs. S9 (a) and (b)) were considered. The cohesive energies of the hcp Ru57 and Ru153 were lower than the regression line obtained from the cohesive energies of the icosahedral fcc Ru-NPs (Fig. S9 (c)). This means that the stable structure of bare Ru-NP is hcp even for the smaller particle sizes when these structures are assumed. Considering that fcc Ru is observed experimentally and that the structure of Ru-NP depends on the precursor and solvent, the ligand is concluded to play an important role in stabilizing the fcc or destabilizing the hcp NP surface. In the previous study, some shapes of uniform crystal structure were treated as a set to prepare a certain number of samples. However, the stability of Ru-NP is sensitive to the shape in spite of the uniform crystal structure. In this study, we considered the various RuNPs with more than 1000 atoms to reveal the shape dependence of the cohesive energy and discussed the detailed elements to decide the structural stability. Considering the shape dependence enables us to understand the intrinsic feature of NP. It was identified that not only surface formation but also twin boundary are important elements to form the fcc Ru-NPs.

The Shape and Size Dependence of the Interatomic Distance of Ru-NPs. We analyzed the optimized structures of Ru-NPs. Figure 5 shows the average interatomic distance between the nearest-neighbor Ru atoms as a function of N–1/3, where the specific values of average interatomic distances were shown in Table S7. The average interatomic distance became shorter with decreasing N. Figure 5 also shows the average interatomic distance between the


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Ru atoms in the surface layer and the interior atoms, between the Ru atoms in the surface layer, and between the interior Ru atoms. The average interatomic distances of the Ru atoms in the surface layer and the inner atoms were longer than those between the surface layer and the inner Ru atoms. The average interatomic distances of the inner Ru atoms were almost constant, while those of the Ru atoms in the surface layer became shorter with decreasing N. The variation of the average interatomic distance between the surface layer and the inner Ru atoms was small compared with that of the Ru atoms in the surface layer. Thus, the interatomic distance of the Ru atoms in the surface layer affects the variation of the average interatomic distance. In the icosahedral fcc Ru-NPs, the average interatomic distance of the nearest-neighbor Ru atoms in the surface layer was longer than that in the inner Ru atoms except for Ru55, which is different from the other types of Ru-NPs. The length of the edges is about 1.05 times as long as that between the edge and the center of the icosahedral structure.63 This relation is retained regardless of the particle size (Figure S10). Thus, the interatomic distance of the Ru atoms in the surface layer is longer than that between the inner Ru atoms, and the average interatomic distances of the icosahedral fcc Ru-NP are longer than those of other types of Ru-NPs.

Analysis of the d-Band Center of Ru-NPs. We analyzed the electronic structure based on the density of states (DOS) for each size and structure. To discuss the difference of the electronic structure depending on the sizes and structures, we focused on the d-band center of the Ru-NPs. Figure 6 shows the d-band center for each size, structure, and site. The atoms except for the surface layer were divided into atoms in the subsurface layer and the inner part. The d-band centers of the inner part in the truncated octahedral fcc and hcp Ru-NPs were about –2.8 eV and –3.0 eV, which are close to the values in the bulk state (Table S3). Thus, the electronic structure of the inner part is similar to the bulk state. On the other hand, the dband center of the atoms in the subsurface layer was lower than that of the inner part. These


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characteristics are common in all types of Ru-NPs. The d-band center was less negative in the order face, edge, and vertex, which corresponds to the decrease in CN. When CN = 9, the dband center became lower with decreasing N, and the d-band center of the icosahedral fcc Ru-NP was higher than those of other types of Ru-NP (Figure S11). The long interatomic distance in the surface layer results in the d-band center upshift together with the narrow dband width.64–67 According to a previous study,64 the icosahedral fcc Ru-NP may show a more negative CO adsorption energy and smaller CO dissociation energy than the other types of Ru-NPs because the interatomic distance in the surface layer is longest in the considered Ru-NPs.

Average Atomic Charge Analysis of Ru-NPs. To understand the details of the electronic structure, we analyzed the atomic charges of Ru-NPs. Figure 7 shows the average atomic charges in the surface and subsurface layers as a function of N–1/3. For all Ru-NPs, the average atomic charges in the surface layer were negative, while those in the subsurface layer were positive. These results indicate the local charge transfer from the Ru atoms in the subsurface layer to the surface layer, as observed for Pd- and Pt-NPs.53 The charge of Ru atoms in the surface layer depends on the site (Figure S12), which is accompanied by the dband center shift, as shown in Figure 6. The cohesion in late transition metals became stronger with decreasing number of electrons, while the energy loss of the bond breaking became smaller with increasing number of electrons.68,69 The charge transfer from the atoms in the subsurface layer to those in the surface layer corresponds to these variations. Thus, the charge transfer makes the NP stable. With decreasing N, the average atomic charges in the subsurface layer increased, while those in the surface layer were almost constant. The difference in the number of atoms between the surface and subsurface layers became larger with decreasing number of atoms in NPs, and the variations were dependent on the shape


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(Table S1). The charge produces an influence on the size dependence of the properties of atoms in the surface layer.

CONCLUSIONS We analyzed the structural stability of fcc and hcp Ru-NPs using DFT calculations. For large particle sizes, the cohesive energy of hcp Ru-NP was the lowest as well as the Ru bulk. With decreasing particle size, the cohesive energy of the icosahedral fcc Ru-NP became closer to that of the hcp Ru-NP. Compared with hcp Ru-NP, the icosahedral fcc Ru-NP having only the fcc {111} facet showed a higher CN even if the number of Ru atoms decreased. In addition, the twin boundary in the icosahedral fcc Ru-NP affected the stability of the cohesive energy because the twin boundary energy per unit area for Ru fcc {111} was negative. The high CN and negative twin boundary are important factors in stabilizing the icosahedral fcc Ru-NP for small particle sizes. However, the lowest cohesive energy is still hcp even if the particle size becomes small. The ligand adsorption may affect the structural stability of Ru-NP. We evaluated the electronic structures of the Ru-NPs based on the d-band center. For the site with low CN, the d-band center of the Ru atoms was high. The Ru atoms with low CN were negatively charged and received more electrons. However, the d-band center of the icosahedral fcc Ru-NP was less negative than those of the other types of Ru-NPs by 0.2–0.4 eV when the Ru atoms with the same CN were compared. The longer interatomic distance of the nearest-neighbor Ru atoms in the surface layer raises the d-band center of the Ru atoms in the surface layer. The icosahedral fcc Ru-NP exhibits not only a high CN and negative twin boundary but also unique geometry and electronic structure characteristics. Both the average CN and the twin boundary are sensitive to the number of atoms, and the electronic structure of the atoms in the surface layer changes with the particle size. Calculations on nanoparticles


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with realistic particle size and shape are necessary to reproduce the observed properties correctly.

ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publication website at DOI:. Details of more information on the Ru nanoparticle model, bulk model, surface model, and twin boundary model.

AUTHOR INFORMATION Corresponding Author *(Y. N.): E-mail: [email protected]. Tel: +81-92-801-6969, Fax: +81-92-8016969., (M. K.): E-mail: [email protected]. Tel: +81-92-801-6968, Fax: +81-92801-6968.

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT The activities of INAMORI Frontier Research Center are supported by Kyocera Corporation. This work was supported by ACCEL, JST and “Advanced Computational Scientific Program” of Research Institute for Information Technology, Kyushu University.

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Rhodium–Gold

Alloy

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(69) Pettifor, D. G. A Physicist’s View of the Energetics of Transition Metals, CALPHAD,

1977, 1, 305–324.


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Tables Table 1. Surface Energies (γ) of Fivefold Twinned Decahedral fcc, Icosahedral fcc, Truncated Octahedral fcc, and hcp Ru-NPsa Deca-fcc N γ 105 287 609 1111

a

0.252 0.210 0.192 0.181

Icosa-fcc N γ 55 147 309 561 923 1415

0.313 0.252 0.223 0.207 0.198 0.191

Truncated-octa fcc N γ 55 201 405 711 807 1289

0.310 0.230 0.212 0.204 0.194 0.191

hcp N

γ

63 167 238 347 625 1021 1557

0.319 0.263 0.253 0.252 0.227 0.215 0.211

The unit of the surface energy is eV/Å2.


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Table 2. Surface Energies (γ) of fcc and hcp Structuresa and Coordination Number (Z) per Atom in the Surface Layer

fcc

γ

Z

(100)

0.186

8

(111)

0.147

9

(0001)

0.164

9

0.184

8

0.213

7

_

hcp

(1010) _

(1120) a The unit of the surface energy is eV/Å2.


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_

_

Table 3. Twin Boundary Energy per Unit Area for fcc {111}, hcp {1012}, and {1011} Stacking a Structure

fcc {111}

_

hcp {1012}

_

hcp {1011}

Number of layers

Twin boundary energy

6L

–1.59 × 10–2

9L

–1.43 × 10–2

12L

–1.29 × 10–2

24L

–1.32 × 10–2

10L

5.66 × 10–2

16L

5.61 × 10–2

20L

5.58 × 10–2

10L

3.43 × 10–2

16L

3.38 × 10–2

20L 3.42 × 10–2 a The unit of the twin boundary energy per unit area is eV/Å2.


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Figures

Figure 1. Models of Ru-NP with fivefold twinned decahedral fcc, icosahedral fcc, truncated octahedral fcc, and hcp structures.


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Figure 2. Cohesive energies of Ru-NP as a function of N–1/3. Red squares, blue diamonds, magenta crosses, and green circles represent fivefold twinned decahedral fcc, icosahedral fcc, truncated octahedral fcc, and hcp structures, respectively. The broken lines represent the linear regression lines. The inset shows an enlarged view at N–1/3 = 0.18–0.24.


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Figure 3. Average CN of Ru-NPs as a function of N–1/3. Red squares, blue diamonds, magenta crosses, and green circles represent fivefold twinned decahedral fcc, icosahedral fcc, truncated octahedral fcc, and hcp structures, respectively. The broken lines represent the linear regression lines.


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Figure 4. Twin boundary models for fcc {111} stacking (Σ3). The original fcc {111} stacking model (Σ1) arranges in ABCABC order.


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Figure 5. Average interatomic distances of the nearest-neighbor Ru atoms in (a) decahedral fcc, (b) icosahedral fcc, (c) truncated octahedral fcc, and (d) hcp Ru-NPs as a function of N−1/3. Black crosses are the average interatomic distance of Ru-NPs. Red squares and blue diamonds represent the average interatomic distances of the nearest neighbor Ru atoms in the surface layer and that between the Ru atoms in the surface layer and the inner Ru atoms, respectively. Green circles are the average interatomic distances between the inner Ru atoms.


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Figure 6. d-Band centers in (a) decahedral fcc, (b) icosahedral fcc, (c) truncated octahedral fcc, and (d) hcp Ru-NPs as a function of N–1/3. Red squares, blue diamonds, green crosses, magenta asterisks, and black circles represent the vertex, edge, face, subsurface layer, and inner part, respectively.


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Figure 7. Average atomic charges of Ru atoms in surface and subsurface layers as a function of N–1/3. Red squares, blue diamonds, magenta crosses, and green circles represent fivefold twinned decahedral fcc, icosahedral fcc, truncated octahedral fcc, and hcp structures, respectively.


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TOC Graphic


33

Structural Stability of Ruthenium Nanoparticles: A Density Functional

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